When we are calculating the loop corrections in our theory (QED, for instance), how does the fact that our electron is in a bound state show itself in the renormalization? Does it matter at all? If so, then why are the predicted values for the Lamb shift, which are calculated in QFT for the electron in Hydrogen, i.e for a bound state, in such close agreement with experiment? Am I missing something here? Thanks

When we are calculating the loop corrections in our theory (QED, for instance), how does the fact that our electron is in a bound state show itself in the renormalization? Does it matter at all? If so, then why are the predicted values for the Lamb shift, which are calculated in QFT for the electron in Hydrogen, i.e for a bound state, in such close agreement with experiment? Am I missing something here? Thanks

The modern approach to QFT in bound states is through the use of effective field theories. For QED bound states, look for NRQED (non relativistic QED). Look for a paper on Lamb shift in NRQED to see the details. The renormalization is done as usual in an effective field theory, the bound state part shows up only in the fact that the asymptotic state is not free particles but a bound state, effectively one must sum up all the ladder Coulomb exchanges to infinity and this corresponds to the asymptotic state being a bound state satisfying Schrodinger's equation with a Coulomb potential. Again, see my paper on Lamb shift in NRQED for details on the Lamb shift calculation in this approach (look also for papers on "potential NRQED").

If your ultimate goal is to study quark bound states, an effective field theory approach is also required. For bound states of heavy quarks, look for NRQCD and potential NRQCD. For bound state of one heavy quark plus light quarks, look for HQET. For light quarks only, one need chiral perturbation theory.